The document discusses various theorems and properties related to triangles, including:
1) The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2) The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
3) Properties relating the lengths of sides and measures of angles in a triangle, such as if sides are unequal then angles will be unequal as well.
All around us, some quantities are constant and others are variable.
For instance, the number of hours in a day is constant, but the number of hours in a daylight in a day is not.
now you will explore more closely certain types of relationships between variables.
The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle.
This document discusses common monomial factoring, which is writing a polynomial as a product of two polynomials where one is a monomial that factors each term. It provides examples of finding the greatest common factor (GCF) of terms in a polynomial and using it to factor the polynomial. Specifically, it factors polynomials like 4m^2 + 10m^4, 6x^4 + 9x^2y + 15x^5y, and 25b^3c^2 - 5b^2c.
Math10 q2 mod3of8_theorems on chords, arcs, central angles and inscribed angl...FahadOdin
The document provides information on a mathematics module for 10th grade covering theorems related to chords, arcs, central angles, and inscribed angles. It includes the development team who created the module, learning objectives, and introduces key concepts and sample proofs involving chords, arcs, central angles, and inscribed angles using two-column proofs.
The document discusses similarity in right triangles. It states that drawing an altitude to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to each other and the original triangle. The altitude also allows one to find missing side lengths using geometric mean formulas. Several examples are given of using these formulas to solve for missing lengths in right triangles.
This document contains a 50 question multiple choice math test covering topics like coordinate geometry, linear equations, functions, and logic. The questions require students to identify properties of linear equations and functions, determine if statements are true or false, identify parts of logical arguments, and choose answers involving math concepts like slope, solutions to inequalities, and properties of shapes. Scripture is included between questions.
The document discusses the key components of a mathematical system:
1. Undefined terms are concepts that cannot be precisely defined, such as points, lines, and planes in geometry.
2. Defined terms have a formal definition using undefined terms or other defined terms, such as line segments, rays, and collinear/coplanar points.
3. Axioms or postulates are statements assumed to be true without proof, which can be used to prove theorems.
4. Theorems are statements that have been formally proven using axioms, postulates and previously proven theorems. The four components are related such that defined terms are defined using undefined terms, axioms are
The document discusses various theorems and properties related to triangles, including:
1) The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2) The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
3) Properties relating the lengths of sides and measures of angles in a triangle, such as if sides are unequal then angles will be unequal as well.
All around us, some quantities are constant and others are variable.
For instance, the number of hours in a day is constant, but the number of hours in a daylight in a day is not.
now you will explore more closely certain types of relationships between variables.
The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle.
This document discusses common monomial factoring, which is writing a polynomial as a product of two polynomials where one is a monomial that factors each term. It provides examples of finding the greatest common factor (GCF) of terms in a polynomial and using it to factor the polynomial. Specifically, it factors polynomials like 4m^2 + 10m^4, 6x^4 + 9x^2y + 15x^5y, and 25b^3c^2 - 5b^2c.
Math10 q2 mod3of8_theorems on chords, arcs, central angles and inscribed angl...FahadOdin
The document provides information on a mathematics module for 10th grade covering theorems related to chords, arcs, central angles, and inscribed angles. It includes the development team who created the module, learning objectives, and introduces key concepts and sample proofs involving chords, arcs, central angles, and inscribed angles using two-column proofs.
The document discusses similarity in right triangles. It states that drawing an altitude to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to each other and the original triangle. The altitude also allows one to find missing side lengths using geometric mean formulas. Several examples are given of using these formulas to solve for missing lengths in right triangles.
This document contains a 50 question multiple choice math test covering topics like coordinate geometry, linear equations, functions, and logic. The questions require students to identify properties of linear equations and functions, determine if statements are true or false, identify parts of logical arguments, and choose answers involving math concepts like slope, solutions to inequalities, and properties of shapes. Scripture is included between questions.
The document discusses the key components of a mathematical system:
1. Undefined terms are concepts that cannot be precisely defined, such as points, lines, and planes in geometry.
2. Defined terms have a formal definition using undefined terms or other defined terms, such as line segments, rays, and collinear/coplanar points.
3. Axioms or postulates are statements assumed to be true without proof, which can be used to prove theorems.
4. Theorems are statements that have been formally proven using axioms, postulates and previously proven theorems. The four components are related such that defined terms are defined using undefined terms, axioms are
The document discusses laws of radicals including the product law of radicals and quotient law of radicals. It provides examples of applying each law such as 6x ∙ 24x = 3√9x ∙ 3√3x^2 and 75 ÷ 3 = 5√25. The learning targets are to derive the laws of radicals and simplify radical expressions using the laws of radicals. Biblical verses from Hebrews 11:6 and Matthew 6:15 about faith are also included.
Final Grade 7 Summative Test (Q3) (1).docxalonajane1
This document contains a third quarter summative test for grade 7 mathematics students in Davao City, Philippines. It consists of 40 multiple choice questions testing students' knowledge of basic geometry topics like angles, lines, polygons, circles, and geometric constructions. The key provides the correct answer for each question.
The document defines the three undefined terms in geometry - point, line, and plane. It explains that a point has no dimensions, a line has one dimension and infinite length, and a plane has two dimensions and infinite length and width. It provides examples of how these terms relate to real-life objects, such as stars being points, a pencil being a line, and a table top being a plane. The document concludes by assigning the reader to draw three real-life objects exemplifying the three terms on a sheet of paper.
Math 8 - Solving Problems Involving Linear FunctionsCarlo Luna
This document is a mathematics lesson on solving problems involving linear functions. It contains 4 practice problems. Problem 1 has students solve for the number of wallets to be sold to make a Php 30 profit and express the profit function in terms of wallets sold. Problem 2 deals with the cost of manufacturing shoes. Problems 3 has students model the number of math problems Cassandrea solves each day as a linear function and use it to determine how many problems she will solve on specific days. The document concludes by providing an asynchronous learning activity for students to complete.
The document defines and describes different types of polygons. It explains that a polygon is a closed figure made of line segments that intersect at endpoints. Polygons are classified as convex or concave depending on whether line extensions of the sides cross the interior or not. Regular polygons are both equilateral (equal sides) and equiangular (equal angles). Examples of different polygons are provided to illustrate these concepts.
The document discusses simple and general annuities. It defines an annuity as a sequence of payments made at regular intervals, such as monthly pension payments or installment loans. A simple annuity has payment and compounding intervals that are equal, while a general annuity has unequal payment and compounding intervals. Examples of simple annuities include car loans with monthly payments and monthly compounding interest. Mortgages are an example of a general annuity, with monthly payments but semi-annual compounding interest. The document also provides formulas for calculating the present and future values of simple and general annuities.
z = kxy
z = -12
z = kxy
z = -84
z = kxy
z = -21
4.
a. Combined means together as a whole.
b. Combined variation is when a quantity varies jointly with respect to the product of two or more variables.
c. The mathematical statement that represents combined variation is a = k(bc) where a varies jointly as b and c multiplied together.
Math 8 - Linear Inequalities in Two VariablesCarlo Luna
This document is a math lesson plan on linear inequalities in two variables taught by Mr. Carlo Justino J. Luna at Malabanias Integrated School in Angeles City. The lesson introduces linear inequalities and their notation, defines them as having two linear expressions separated by symbols like greater than and less than, and shows examples of inequalities in two variables. It then discusses how to determine if an ordered pair is a solution by substituting into the inequality. Finally, it explains how to graph linear inequalities in two variables by first rewriting them as equations and then plotting intercepts and shading the appropriate region based on a test point.
The document is a science review test with multiple choice questions covering various topics in science including:
- The human circulatory system and blood flow through the heart and lungs
- Components and functions of the nervous system including brain, spinal cord, and nerves
- Food chains and webs showing trophic levels and energy transfer between organisms
- Biogeochemical cycles like oxygen-carbon dioxide and water cycles
- Energy transformations and forms of energy
- Forces and laws of motion
- Theories of the origin and structure of the universe, Earth, and life
The document defines important terms related to permutation and combination, including factorial, fundamental principle of counting, and different types of permutations and combinations. It provides examples of calculating permutations with and without repetition, as well as combinations with and without repetition. Formulas are given for each case. Permutations refer to arrangements where order matters, while combinations are arrangements where order does not matter.
This document provides a detailed lesson plan for a 9th grade mathematics class on parallelograms and triangle similarity. The lesson plan includes objectives, content, learning resources, procedures, and evaluation. The procedures involve a jigsaw activity to identify shapes, a discussion of properties of quadrilaterals, an activity to determine if statements about quadrilaterals are true or false, drawing and describing different quadrilaterals, and making a short jingle about what was learned. The evaluation consists of individually answering questions to draw and identify various quadrilaterals and determine if statements about them are true or false.
This powerpoint presentation discusses or talks about the topic or lesson: Laws of Exponents. It also discusses and explains the rules, concepts, steps and examples of Laws of Exponents.
The document discusses several theorems related to triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
This document provides objectives and activities for a lesson on organizing and presenting data using appropriate graphs. The objectives are to enumerate different graph types, organize data using suitable graphs, and apply the lesson through a survey. Example graphs shown include histograms, pie charts, bar graphs and line graphs. Descriptions of each graph type are provided. Students are instructed to conduct a survey to identify the most popular teacher and organize the results in a graph. Additional practice with graphing real data from websites is suggested.
This document provides information about a mathematics module on similarity for grade 9 learners. It was collaboratively developed by educators from various educational institutions in the Philippines. The module aims to teach learners about proportions, similarity of polygons, conditions for similarity of triangles using various theorems, applying similarity to solve real-world problems involving proportions and similarity. It includes a module map, pre-assessment questions to gauge learners' prior knowledge, and covers topics like proportions, similarity of polygons and triangles, and applying similarity concepts to solve problems.
Powerpoint on K-12 Mathematics Grade 7 Q1 (Fundamental Operations of Integer...Franz Jeremiah Ü Ibay
This document provides information on addition, subtraction, multiplication, and division of integers. It begins by explaining that when adding or multiplying integers with the same sign, you keep the same sign, and with different signs, the result is negative. Examples are provided to illustrate addition, subtraction, multiplication, and division of integers. The document then discusses properties of integer operations like closure, commutativity, associativity, distributivity, identity, and inverses. Activities are included for students to practice integer operations.
Addition and Subtraction Property of EqualitySonarin Cruz
This document discusses the addition and subtraction properties of equality. It explains that for any real numbers a, b, and c, if a + c = b + c, then the addition property of equality holds. Similarly, if a - c = b - c, then the subtraction property of equality holds. It provides examples of using these properties to solve equations by adding or subtracting the same quantity to both sides of an equation. The document encourages working through practice problems to determine the solution of equations.
This document contains a daily lesson log for an 8th grade mathematics class. The lesson focuses on linear equations in two variables and systems of linear equations over four class periods. Key points covered include graphing and finding equations of lines given various parameters, solving systems of linear equations by elimination, and applying these concepts to word problems. Formative assessments are built into the lesson procedures to evaluate student understanding.
This document contains a daily lesson log/plan for a mathematics class covering polynomial equations. The lesson plan outlines objectives, topics, materials, procedures and activities for teaching polynomial equations over the course of a week. Key points covered include identifying polynomial equations, writing polynomial terms in descending order, determining the degree and coefficients of polynomials, solving polynomial equations using factoring and synthetic division, and writing polynomial equations given their roots. Assessment activities include writing examples of solving polynomial equations and problems involving polynomials.
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
The document discusses laws of radicals including the product law of radicals and quotient law of radicals. It provides examples of applying each law such as 6x ∙ 24x = 3√9x ∙ 3√3x^2 and 75 ÷ 3 = 5√25. The learning targets are to derive the laws of radicals and simplify radical expressions using the laws of radicals. Biblical verses from Hebrews 11:6 and Matthew 6:15 about faith are also included.
Final Grade 7 Summative Test (Q3) (1).docxalonajane1
This document contains a third quarter summative test for grade 7 mathematics students in Davao City, Philippines. It consists of 40 multiple choice questions testing students' knowledge of basic geometry topics like angles, lines, polygons, circles, and geometric constructions. The key provides the correct answer for each question.
The document defines the three undefined terms in geometry - point, line, and plane. It explains that a point has no dimensions, a line has one dimension and infinite length, and a plane has two dimensions and infinite length and width. It provides examples of how these terms relate to real-life objects, such as stars being points, a pencil being a line, and a table top being a plane. The document concludes by assigning the reader to draw three real-life objects exemplifying the three terms on a sheet of paper.
Math 8 - Solving Problems Involving Linear FunctionsCarlo Luna
This document is a mathematics lesson on solving problems involving linear functions. It contains 4 practice problems. Problem 1 has students solve for the number of wallets to be sold to make a Php 30 profit and express the profit function in terms of wallets sold. Problem 2 deals with the cost of manufacturing shoes. Problems 3 has students model the number of math problems Cassandrea solves each day as a linear function and use it to determine how many problems she will solve on specific days. The document concludes by providing an asynchronous learning activity for students to complete.
The document defines and describes different types of polygons. It explains that a polygon is a closed figure made of line segments that intersect at endpoints. Polygons are classified as convex or concave depending on whether line extensions of the sides cross the interior or not. Regular polygons are both equilateral (equal sides) and equiangular (equal angles). Examples of different polygons are provided to illustrate these concepts.
The document discusses simple and general annuities. It defines an annuity as a sequence of payments made at regular intervals, such as monthly pension payments or installment loans. A simple annuity has payment and compounding intervals that are equal, while a general annuity has unequal payment and compounding intervals. Examples of simple annuities include car loans with monthly payments and monthly compounding interest. Mortgages are an example of a general annuity, with monthly payments but semi-annual compounding interest. The document also provides formulas for calculating the present and future values of simple and general annuities.
z = kxy
z = -12
z = kxy
z = -84
z = kxy
z = -21
4.
a. Combined means together as a whole.
b. Combined variation is when a quantity varies jointly with respect to the product of two or more variables.
c. The mathematical statement that represents combined variation is a = k(bc) where a varies jointly as b and c multiplied together.
Math 8 - Linear Inequalities in Two VariablesCarlo Luna
This document is a math lesson plan on linear inequalities in two variables taught by Mr. Carlo Justino J. Luna at Malabanias Integrated School in Angeles City. The lesson introduces linear inequalities and their notation, defines them as having two linear expressions separated by symbols like greater than and less than, and shows examples of inequalities in two variables. It then discusses how to determine if an ordered pair is a solution by substituting into the inequality. Finally, it explains how to graph linear inequalities in two variables by first rewriting them as equations and then plotting intercepts and shading the appropriate region based on a test point.
The document is a science review test with multiple choice questions covering various topics in science including:
- The human circulatory system and blood flow through the heart and lungs
- Components and functions of the nervous system including brain, spinal cord, and nerves
- Food chains and webs showing trophic levels and energy transfer between organisms
- Biogeochemical cycles like oxygen-carbon dioxide and water cycles
- Energy transformations and forms of energy
- Forces and laws of motion
- Theories of the origin and structure of the universe, Earth, and life
The document defines important terms related to permutation and combination, including factorial, fundamental principle of counting, and different types of permutations and combinations. It provides examples of calculating permutations with and without repetition, as well as combinations with and without repetition. Formulas are given for each case. Permutations refer to arrangements where order matters, while combinations are arrangements where order does not matter.
This document provides a detailed lesson plan for a 9th grade mathematics class on parallelograms and triangle similarity. The lesson plan includes objectives, content, learning resources, procedures, and evaluation. The procedures involve a jigsaw activity to identify shapes, a discussion of properties of quadrilaterals, an activity to determine if statements about quadrilaterals are true or false, drawing and describing different quadrilaterals, and making a short jingle about what was learned. The evaluation consists of individually answering questions to draw and identify various quadrilaterals and determine if statements about them are true or false.
This powerpoint presentation discusses or talks about the topic or lesson: Laws of Exponents. It also discusses and explains the rules, concepts, steps and examples of Laws of Exponents.
The document discusses several theorems related to triangles:
1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference.
2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles.
3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.
This document provides objectives and activities for a lesson on organizing and presenting data using appropriate graphs. The objectives are to enumerate different graph types, organize data using suitable graphs, and apply the lesson through a survey. Example graphs shown include histograms, pie charts, bar graphs and line graphs. Descriptions of each graph type are provided. Students are instructed to conduct a survey to identify the most popular teacher and organize the results in a graph. Additional practice with graphing real data from websites is suggested.
This document provides information about a mathematics module on similarity for grade 9 learners. It was collaboratively developed by educators from various educational institutions in the Philippines. The module aims to teach learners about proportions, similarity of polygons, conditions for similarity of triangles using various theorems, applying similarity to solve real-world problems involving proportions and similarity. It includes a module map, pre-assessment questions to gauge learners' prior knowledge, and covers topics like proportions, similarity of polygons and triangles, and applying similarity concepts to solve problems.
Powerpoint on K-12 Mathematics Grade 7 Q1 (Fundamental Operations of Integer...Franz Jeremiah Ü Ibay
This document provides information on addition, subtraction, multiplication, and division of integers. It begins by explaining that when adding or multiplying integers with the same sign, you keep the same sign, and with different signs, the result is negative. Examples are provided to illustrate addition, subtraction, multiplication, and division of integers. The document then discusses properties of integer operations like closure, commutativity, associativity, distributivity, identity, and inverses. Activities are included for students to practice integer operations.
Addition and Subtraction Property of EqualitySonarin Cruz
This document discusses the addition and subtraction properties of equality. It explains that for any real numbers a, b, and c, if a + c = b + c, then the addition property of equality holds. Similarly, if a - c = b - c, then the subtraction property of equality holds. It provides examples of using these properties to solve equations by adding or subtracting the same quantity to both sides of an equation. The document encourages working through practice problems to determine the solution of equations.
This document contains a daily lesson log for an 8th grade mathematics class. The lesson focuses on linear equations in two variables and systems of linear equations over four class periods. Key points covered include graphing and finding equations of lines given various parameters, solving systems of linear equations by elimination, and applying these concepts to word problems. Formative assessments are built into the lesson procedures to evaluate student understanding.
This document contains a daily lesson log/plan for a mathematics class covering polynomial equations. The lesson plan outlines objectives, topics, materials, procedures and activities for teaching polynomial equations over the course of a week. Key points covered include identifying polynomial equations, writing polynomial terms in descending order, determining the degree and coefficients of polynomials, solving polynomial equations using factoring and synthetic division, and writing polynomial equations given their roots. Assessment activities include writing examples of solving polynomial equations and problems involving polynomials.
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
This document discusses polynomials. It defines a polynomial as an expression constructed from variables and constants using addition, subtraction, multiplication, and exponents. Polynomials appear in many areas of mathematics and science. The document then discusses the terms, coefficients, and different types of polynomials including linear, quadratic, cubic, and constant polynomials. It also discusses the degree and maximum number of zeroes of each type of polynomial. Finally, it discusses using the division algorithm to find quotients, remainders, and factors of polynomials.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
Mathematics power point presenttation on the topicMeghansh Gautam
This document provides an overview of mathematics and different types of numbers. It discusses what mathematics is, polynomials, algebraic identities, and various number systems including natural numbers, integers, rational numbers, real numbers, complex numbers, and computable numbers. It also briefly discusses the history of numbers, mentioning that tally marks found on bones and artifacts may be some of the earliest forms of counting and record keeping.
The document provides information about the Remainder Theorem including:
1. The objectives are to prove the Remainder Theorem and evaluate polynomials using it.
2. It reviews the terms used in polynomial long division and provides an example.
3. The lesson will have students work in groups to solve remainder problems in a race activity.
This document provides a guide for 8th grade mathematics teaching support for the second semester of 2021 from the INEM JORGE ISAACS Educational Institution. It includes the names of 13 teachers, details that the guide must be submitted by August 27 according to the schedule, and outlines standards and basic learning rights related to algebraic expressions, equations, and graphs. It also provides details on levels of performance, prior knowledge activities to be completed in June and July on polynomials, and explanations of polynomial operations like addition, subtraction, multiplication, and division with examples.
1. Mathematics can be thought of as a language with its own symbols, grammar, and structure for conveying precise ideas. It uses symbols like numbers, variables, and operations in place of words.
2. Some key aspects of mathematical language include its precision, conciseness, and power to solve complex problems using symbolic representation. Mathematics has its own grammar rules for using symbols in expressions, equations, and different types of mathematical statements.
3. There are different types of mathematical statements, including universal statements that hold for all cases, existential statements that hold for at least one case, conditional statements of the form "if P then Q", and universal conditional statements that are both universal and conditional.
1. Turn on your camera and mute your microphone only when instructed to do so. Listen carefully and speak one at a time without using the conversation box.
2. The document discusses factoring polynomials by finding the greatest common factor (GCF) of terms. It provides examples of factoring polynomials that have a common monomial factor like 4x^2 and 6x.
3. Students are instructed on how to find the GCF, which is the largest numerical coefficient and variable factor that is common between all terms. They are provided examples of factoring polynomials using the distributive property.
The document outlines the daily lesson log for a Grade 10 mathematics class. It includes 4 sessions on the topic of polynomial functions. The objectives are to illustrate, find intercepts of, and graph polynomial functions. Examples are provided for identifying key features of polynomials like degree, leading coefficient, constant term, and finding x- and y-intercepts by setting the polynomial equal to 0 or using the factored form. The behavior of graphs is also discussed based on whether the degree is odd or even, and the sign of the leading coefficient.
- A rational function is a function that can be written as the ratio of two polynomial functions. Graphs of rational functions may have breaks in continuity where there is a vertical asymptote or point discontinuity.
- Mathematician Maria Gaetana Agnesi discussed the "curve of Agnesi", the equation x2/y = a2(a - y), which can be expressed as a rational function y = a2/x2 + a2.
- A polynomial is the sum of monomial terms. The degree of a polynomial is the degree of the highest-degree term. The FOIL method can be used to multiply binomials by distributing and combining like terms.
- A rational function is a function that can be written as the ratio of two polynomial functions. Graphs of rational functions may have breaks in continuity where there is a vertical asymptote or point of discontinuity.
- Mathematician Maria Gaetana Agnesi discussed the "curve of Agnesi", the equation x2/y = a2(a - y), which can be written as a rational function y = a2/x2 + a2.
- A polynomial is the sum of monomial terms. The degree of a polynomial is the degree of the highest-degree term. The FOIL method can be used to multiply binomials by distributing and combining like terms.
The document discusses polynomials. It defines polynomials as expressions constructed from variables and constants using addition, subtraction, multiplication, and non-negative integer exponents. It provides examples of polynomials and non-polynomial expressions. It also discusses the degrees of terms and polynomials, and how polynomials can be added or multiplied by distributing terms. The document also covers monomials, binomials, trinomials, and the factor theorem.
This document is a project report submitted by a student for a Bachelor of Science degree in Mathematics. It provides an introduction to the field of topology. The report includes a title page, certificate, declaration, acknowledgements, table of contents, and several chapters. The introduction defines topology and provides some examples. It states that the project aims to provide a thorough grounding in general topology.
The document outlines a lesson plan on determining the inverse of one-to-one functions. The objectives are for students to understand inverse functions and apply them to solve real-life problems. The content covers defining the inverse of a one-to-one function, determining the inverse by interchanging variables and solving for y in terms of x. Examples of finding specific inverses are provided. The lesson concludes with an evaluation where students find the inverses of several functions.
Polynomials are mathematical expressions constructed from variables and constants using addition, subtraction, multiplication, and exponents of whole numbers. They appear in many areas of mathematics and science. Polynomials can be used to form equations that model problems in various domains. They also define polynomial functions that are used in fields like physics, chemistry, economics, and social sciences. Polynomials are classified based on their degree, with linear polynomials having degree 1 and quadratic polynomials degree 2. The maximum number of zeroes a polynomial can have is equal to its degree.
1) Students will complete a two-part writing assignment on polynomial functions.
2) In part one, students must factor a polynomial, find all its roots, sketch its graph, and explain its end behavior and behavior at roots using formal algebra.
3) In part two, students must construct a polynomial in standard form given its roots, explaining the theorems used to find remaining roots and showing calculation steps.
This document discusses polynomial functions. It defines polynomial expressions and equations. The objectives are to recall concepts of polynomial expressions, illustrate polynomial functions, and find the degree and leading term. It provides examples of determining if something is a polynomial expression and examples of writing polynomial functions in standard form. It also defines the degree, leading coefficient, and constant term of a polynomial function.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
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2. Activity1: Charade/Draw on board with blindfold.
Mechanics:
1. A representative from the group will draw a piece of paper from a
box. The representative will sketch on the board (with blindfold) or act
out (charade) what is written on the paper.
2. Other group members will guess the word within a time limit. If the
group will not be able to answer, then the other group may steal the
chance to answer.
3. 1. Engineers use polynomials to graph
the curves of roller coasters. Since
polynomials are used to describe curves
of various types, people use them in the
real world to graph curves. For example,
roller coaster designers may use
polynomials to describe the curves in
their rides.
Examples of which polynomials can be applied in real life.
4. 2. Polynomials can also be used to
model different situations, like in the
stock market to see how prices will vary
over time. Business people also use
polynomials to model markets, as in to
see how raising the price of a good will
affect its sales. In economics they are
used to conduct cost analysis. (used to
interpret and forecast market trends)
Examples of which polynomials can be applied in real life.
5. 3. Polynomials are used in physics to
describe the trajectory of projectiles.
Projectile motion is a form of motion
where an object moves in parabolic path;
the path that the object follows is called
its trajectory.(Missiles)
Examples of which polynomials can be applied in real life.
6. 4. Science Careers Physical and social scientists, including
archaeologists, astronomers, meteorologists, chemists and physicists,
need to use polynomials in their jobs. Key scientific formulas, including
gravity equations, feature polynomial expressions. These algebraic
equations help scientists to measure relationships between
characteristics such as force, mass and acceleration. Astronomers
use polynomials to help in finding new stars and planets and
calculating their distance from Earth, their temperature and other
features, according to schoolfor-champions.com
Examples of which polynomials can be applied in real life.
8. Students is expected to:
1. identify polynomial functions,
2. determines the degree, leading term,
leading coefficient, constant term in a given
polynomial function,
3. relate the topic in real life situation.
OBJECTIVES:
9. Activity 2: Fill in the Blanks
Mechanics: The students will
review the previous topic on how to
determine a polynomial function by
competing the given
statement/definition.
10. 1.The word polynomial is made of two words, ____ and ____,
meaning many terms.
2.A _____is made up of terms and each term has a coefficient
(numerical coefficient), variable (literal coefficient) and
____while an ____is a sentence with a minimum of two
numbers and at least one math operation in it.
3.Monomial, binomial and trinomial are _____ of polynomial.
4.The _____ of a polynomial refers to the_____ degree among
the degrees of the terms in the polynomial.
5.The degree of a _____in a polynomial refers to the exponent of
variable/literal coefficient.
11. A polynomial _____ is a function that
can be expressed in the form of a
polynomial. Generally, it is represented
as ____.
Bonus item: Fill in the blank.
12. A polynomial function is a function that can be
expressed in the form of a polynomial. Generally, it
is represented as P(x).
a function defined by:
P(x) = an xn + an-1 xn-1+.……….…+a2 x2 + a1 x + a0
Where a0 a1 , …, an-1, an, are real numbers, an ≠ 0, and
n is nonnegative integer.
LET’S EXPORE!
13. LET’S EXPORE!
A polynomial function
is a function that can
be expressed in the
form of a polynomial.
Generally, it is
represented as P(x).
14. Which of the following are polynomial function? Not
polynomial function? Determine the degree, leading
term, leading coefficient, and constant term.
1)P(x)= 3x2 + x – 1
2)P(x)= 2x4 + x2 – 3x + 𝟓𝒙1/2
3)P(x)= – 10 + x – 3x3 – x5
4)𝐏(𝐱) = 𝟑𝒙𝟔 + x4 – 3x2 + 4
EXAMPLE:
15. Activity # 3: On your Own.
A.Which of the following are polynomial function? Not polynomial
function? Determine the degree of the polynomial, leading term,
leading coefficient, constant term and write the polynomials in
standard form.
1)P(x)= 2x3 + x4 – x + 5
2)P(x)= x4 + x2 – 3x + 𝒙𝟑 - 2
3)P(x)= 8 + x2 – 3x3 + 2x-4
4)𝐏(𝐱) = + x4 – 3x2 + 41/3
5)𝐏 𝐱 =
𝟏
𝒙
+ x4 – 3x2 + 1
16. Let’s Wrap Up!
1.) A ________ of degree n in x is an algebraic expression that contains a
specific number of terms each of which is of the form axn, where a a is a
real number and n is a whole number.
2.) ________ of Polynomials
1. Monomial – polynomial with one term
2. Binomial – polynomial with two terms
3. Trinomial – polynomial with three terms
4. Multinomial (polynomial)
3.) The degree of a ________ in a polynomial in x refers to the exponent of
x.
17. Let’s Wrap Up!
4.) The _________ of a polynomial refers to the highest
degree among the degrees of the terms in the
polynomial.
5.) Polynomial _________ - is simply a polynomial that has
been set equal to zero in an equation.
6.) A __________ is a function that can be expressed in the
form of a polynomial. Generally, it is represented as P(x). a
function defined by P(x) = an xn + an-1 xn-1+.……….…+a2 x2 +
a1 x + a0.
18. LET’S EVALUATE!
Choose the letter of the correct answer. Then write it on a separate sheet of
paper.
1.Which of the following is an example of polynomial?
a.term c. degree
b.binomial d. algebraic expression
2. It is simply a polynomial that has been set equal to zero
in an equation.
a.Polynomial c. Polynomial function
b.Polynomial expression d. Polynomial equation
19. LET’S EVALUATE!
Choose the letter of the correct answer. Then write it on a separate sheet of
paper.
3. What do you mean by polynomial function?
a.It refers to the highest degree among the degrees of the terms in
the polynomial.
b.It refers to the exponent of the variable.
c. It can be expressed in the form of a polynomial. Generally, it is
represented as P(x). a function defined by P(x) = an xn + an-1 xn-
1+.……….…+a2 x2 + a1 x + a0.
d.an algebraic expression that contains a specific number of terms
each of which is of the form axn, where a a is a real number and
n is a whole number.
20. LET’S EVALUATE!
Choose the letter of the correct answer. Then write it on a separate sheet of
paper.
4. Which of the following is a polynomials function?
a.P(x) = 3x-2 + x – 1 b. P(x) = 2x4 + x2 – 3x + 𝟓𝒙1/2
c. – 10 + x – 3x3 – x5 d. P(x) = 3𝑥6 + x4 – 3x2 + 4
21. LET’S EVALUATE!
Choose the letter of the correct answer. Then write it on a separate sheet of
paper.
5. Determine the degree of the polynomial, leading term, the
leading coefficient and constant term of polynomial function P(x)=
x3 + 3x2 – 4x + 2𝒙4
a. degree of the polynomial = 3, leading term= x3 , leading coefficient= 1,
constant term= 2
b. degree of the polynomial = 3, leading term= x3 , leading coefficient= 2,
constant term= 0
c. degree of the polynomial = 4, leading term= 2x4 , leading coefficient= 1,
constant term= 2
d. degree of the polynomial = 4, leading term= 2x4 , leading coefficient= 2,
constant term= 0